Stabilized tetrahedral elements for crystal plasticity finite element analysis overcoming volumetric locking

Abstract

Image-based CPFE modeling involves computer generation of virtual polycrystalline microstructures from experimental data, followed by discretization into finite element meshes. Discretization is commonly accomplished using three-dimensional four-node tetrahedral or TET4 elements, which conform to the complex geometries. It has been commonly observed that TET4 elements suffer from severe volumetric locking when simulating deformation of incompressible or nearly incompressible materials. This paper develops and examines three locking-free stabilized finite element formulations in the context of crystal plasticity finite element analysis. They include a node-based uniform strain (NUS) element, a locally integrated B-bar (LIB) based element and a F-bar patch (FP) based element. All three formulations are based on the partitioning of TET4 element meshes and integrating over patches to obtain favorable incompressibility constraint ratios without adding large degrees of freedom. The results show that NUS formulation introduces unstable spurious energy modes, while the LIB and FP elements stabilize the solutions and are preferred for reliable CPFE analysis. The FP element is found to be computationally efficient over the LIB element.

Appendix: Evaluation of the tangent stiffness matrix

In this appendix, the tangent stiffness tensor $\mathbf {C}^{t}$ in equation (8) is derived from the crystal plasticity constitutive model. $\mathbf {C}^{t}$ at time t is expressed as:

$$\begin{aligned} \mathbf {C}^{t}=\frac{1}{det\mathbf {F}_0^t} \, \left( \mathbf {F}_0^t {\otimes } \mathbf {F}_0^t\,\right) :\mathbf {C}^{0}:\left( \mathbf {F}_0^t {\otimes } \mathbf {F}_0^t\right) \end{aligned}$$

(63)

where $\mathbf {C}^{0}$ has been derived in [55] as:

$$\begin{aligned} \mathbf {C}^{0}= & {} \frac{\partial \,\mathbf {S}_{0}^{t}}{\partial \,\mathbf {E}_{0}^{t}}=\left( det\mathbf {F}^{p}\right) \left( \mathbf {F^{p}}\underline{\otimes }\mathbf {F}^{p}\right) ^{-1} \nonumber \\&:\left\{ \frac{\partial \mathbf {S}^{*}}{\partial \mathbf {E}}+\left[ \mathbf {S}^{*}\otimes \mathbf {F}^{p^{-T}}-\left( det\mathbf {F}^{p}\right) ^{-1}\left[ \mathbf {I}\underline{\otimes }\left( \mathbf {S}\mathbf {F}^{p^{T}}\right) ^{T}\right. \right. \right. \nonumber \\&\left. \left. \left. +\left( \mathbf {F}^{p}\mathbf {S}\right) \bar{\otimes }\mathbf {I}\right] \right] :\frac{\partial \mathbf {F}^{p}}{\partial \mathbf {E}}\right\} \end{aligned}$$

(64)

with

$$\begin{aligned}&\mathbf {S}^{*}=(det\mathbf {F}^{p})^{-1}\mathbf {F}^{p}\mathbf {S}\mathbf {F}^{p^{T}} \end{aligned}$$

(65a)

$$\begin{aligned}&\frac{\partial \mathbf {S}^{*}}{\partial \mathbf {E}}=\left[ \mathbf {I}\underline{\otimes }\mathbf {I}+\sum _{\alpha }^{N_\mathrm{slip}}\left( C^{\alpha }\otimes \frac{\partial \triangle \gamma ^{\alpha }}{\partial \mathbf {S}^{*}}\right) \right] ^{-1}\nonumber \\ {}&\qquad \quad \left[ \mathbf {A}^{\alpha }-\sum _{\alpha }^{N_\mathrm{slip}}\triangle \gamma ^{\alpha }\mathbf {B}^{\alpha }\right] \end{aligned}$$

(65b)

$$\begin{aligned}&\mathbf {A}=\mathbf {C}^{e}:\left( \mathbf {F}^{p^{-1}}\underline{\otimes } \,\mathbf {F}^{p^{-1}}\right) \end{aligned}$$

(65c)

$$\begin{aligned}&\mathbf {B}^{\alpha }=\mathbf {C}^{e}:\left[ \mathbf {F}^{p^{-T}}\underline{\otimes }\left( \mathbf {F}^{p^{-1}}\mathbf {s}_{0}^{\alpha }\right) ^{T}+\left( \mathbf {F}^{p^{-1}}\mathbf {s}_{0}^{\alpha }\right) ^{T}\underline{\otimes } \,\mathbf {F}^{p^{-T}}\right] \end{aligned}$$

(65d)

$$\begin{aligned}&\frac{\partial \mathbf {F}^{p}}{\partial \mathbf {E}}=\sum _{\alpha }^{N_\mathrm{slip}}\left( \mathbf {s}_{0}^{\alpha }\mathbf {F}^{p}\right) \otimes \left( \frac{\partial \triangle \gamma ^{\alpha }}{\partial \mathbf {S}^{*}}\frac{\partial \mathbf {S}^{*}}{\partial \mathbf {E}}\right) \end{aligned}$$

(65e)

The lower and upper tensor product operators $\underline{\otimes }$ and $\bar{\otimes }$ are defined as $\left( A\underline{\otimes }B\right) _{ijkl}=A_{ik}B_{jl}$ and $\left( A\bar{\otimes }B\right) _{ijkl}=A_{il}B_{jk}$. $\mathbf {C}^{t}$ is a function of path-dependent state variables, i.e. $\mathbf {C}^{t}=\mathbf {C}^{t}\left( \mathbf {F}^{t} ,{\mathbf {F}^{p}}^{t}, \dot{\gamma }^{t}, \ldots \right)$. The time-integration algorithm developed in [34] is implemented here to incrementally update state variables. The fourth order tensor $\mathbf {C}^{t}$ is written as a $6\times 6$ matrix, using the property of major symmetry, for implementation to finite element weak form.